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Properties

Label 1-1441-1441.736-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.661 - 0.749i$
Analytic cond. $154.856$
Root an. cond. $154.856$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.215 − 0.976i)2-s + (−0.998 − 0.0483i)3-s + (−0.906 + 0.421i)4-s + (0.836 − 0.548i)5-s + (0.168 + 0.985i)6-s + (0.443 + 0.896i)7-s + (0.607 + 0.794i)8-s + (0.995 + 0.0965i)9-s + (−0.715 − 0.698i)10-s + (0.926 − 0.377i)12-s + (−0.715 + 0.698i)13-s + (0.779 − 0.626i)14-s + (−0.861 + 0.506i)15-s + (0.644 − 0.764i)16-s + (0.0724 − 0.997i)17-s + (−0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (−0.215 − 0.976i)2-s + (−0.998 − 0.0483i)3-s + (−0.906 + 0.421i)4-s + (0.836 − 0.548i)5-s + (0.168 + 0.985i)6-s + (0.443 + 0.896i)7-s + (0.607 + 0.794i)8-s + (0.995 + 0.0965i)9-s + (−0.715 − 0.698i)10-s + (0.926 − 0.377i)12-s + (−0.715 + 0.698i)13-s + (0.779 − 0.626i)14-s + (−0.861 + 0.506i)15-s + (0.644 − 0.764i)16-s + (0.0724 − 0.997i)17-s + (−0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (736, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4937708847 - 1.094495358i\)
\(L(\frac12)\) \(\approx\) \(0.4937708847 - 1.094495358i\)
\(L(1)\) \(\approx\) \(0.6947364735 - 0.3718850102i\)
\(L(1)\) \(\approx\) \(0.6947364735 - 0.3718850102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.215 - 0.976i)T \)
3 \( 1 + (-0.998 - 0.0483i)T \)
5 \( 1 + (0.836 - 0.548i)T \)
7 \( 1 + (0.443 + 0.896i)T \)
13 \( 1 + (-0.715 + 0.698i)T \)
17 \( 1 + (0.0724 - 0.997i)T \)
19 \( 1 + (0.970 + 0.239i)T \)
23 \( 1 + (0.958 - 0.285i)T \)
29 \( 1 + (-0.995 + 0.0965i)T \)
31 \( 1 + (0.0241 + 0.999i)T \)
37 \( 1 + (0.485 - 0.873i)T \)
41 \( 1 + (-0.644 - 0.764i)T \)
43 \( 1 + (0.262 + 0.964i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.527 - 0.849i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.262 + 0.964i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.354 - 0.935i)T \)
83 \( 1 + (0.681 - 0.732i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.989 + 0.144i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.01688413385467531133832286004, −19.99264025630616115268614578528, −18.89147028266485326623428860622, −18.28023878802305248816874240455, −17.54119855728996567322964273908, −17.00094767399752532158599337970, −16.759680579554669328458594968571, −15.38191311255357187256705330343, −15.003729996905634800148951713389, −14.04301992328769956218047521615, −13.28515955052487996929891772430, −12.709439697576367791101653751884, −11.31235984379252386018344732781, −10.70658577274168910595911051940, −9.904420773147408172534096761664, −9.47861279822717434041246592149, −7.99696739849050865714520909589, −7.35575763657982385428209079561, −6.68277002844184084588815864713, −5.8367890648657814267912518113, −5.215758199116212791571741153979, −4.441467404988080935321703591925, −3.31744955905252849599588740084, −1.64323407579617858099797511699, −0.83439757783473233538639181736, 0.37019313565231291565579714594, 1.43189877161598690246330340366, 2.061674024588006776965259544529, 3.124757139726303849414886464233, 4.59297333277781196440158606751, 5.069629281036982945139840594364, 5.65267613286857488963799367749, 6.88012635182962792432336268034, 7.85182318724692304786188031491, 9.17115503137319298950827282429, 9.32795672293249444353902545498, 10.27551801561086139305848068824, 11.19740971284283001501391254713, 11.82691016274797828061982025155, 12.391191953408770149393814155211, 13.091786586866611352292350695090, 13.97294144499123574048542360736, 14.77416427007170319900798588832, 16.16601311632329825920333420071, 16.621624805086646016318882404112, 17.53714540853310741860718388330, 18.01642887513854787472691705726, 18.60559236750520929074913369260, 19.38995071696128359032170182136, 20.562626053467299441121934796609

Graph of the $Z$-function along the critical line